Almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres

TL;DR

This paper investigates almost complex and para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres, revealing their non-integrability and contrasting their properties with those on the standard Riemann sphere.

Contribution

It introduces and analyzes Cayley structures on pseudo-Riemannian spheres, showing their non-integrability and existence of integrable structures unlike the classical case.

Findings

Cayley structures are non-integrable.

Existence of integrable complex and para-complex structures on pseudospheres.

Basic geometric characteristics are explicitly calculated.

Abstract

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculated. In contrast to the usual Riemann sphere $\mathbb{S}^6$, there exist (integrable) complex structures and para-complex structures on the pseudospheres under consideration.